Examples and Problems in Mathematical Statistics

CHAPTER 7

Large Sample Theory for Estimation and Testing

PART I: THEORY

We have seen in the previous chapters several examples in which the exact sampling distribution of an estimator or of a test statistic is difficult to obtain analytically. Large samples yield approximations, called asymptotic approximations, which are easy to derive, and whose error decreases to zero as the sample size grows. In this chapter, we discuss asymptotic properties of estimators and of test statistics, such as consistency, asymptotic normality, and asymptotic efficiency. In Chapter 1, we presented results from probability theory, which are necessary for the development of the theory of asymptotic inference. Section 7.1 is devoted to the concept of consistency of estimators and test statistics. Section 7.2 presents conditions for the strong consistency of the maximum likelihood estimator (MLE). Section 7.3 is devoted to the asymptotic normality of MLEs and discusses the notion of best asymptotically normal (BAN) estimators. In Section 7.4, we discuss second and higher order efficiency. In Section 7.5, we present asymptotic confidence intervals. Section 7.6 is devoted to Edgeworth and saddlepoint approximations to the distribution of the MLE, in the one–parameter exponential case. Section 7.7 is devoted to the theory of asymptotically efficient test statistics. Section 7.8 discusses the Pitman’s asymptotic efficiency of tests.

7.1 CONSISTENCY OF ESTIMATORS AND TESTS

Consistency of an estimator is a ...